Question

Use R programming to solve

Q2. A matrix operator H(G; k) on a pxp symmetric matrix G (iy)- with a positive integer parameter k (k < p) yields another p×p symmetric matrix H = (hij 1 with i=k,j = k; (a) Use one single loop to construct the function H(G; k) in R (b) Generate a random matrix X of dimension 7x5, each element of which is id from N(0,1). Use the function H(G; k constructed in (a) to compute H(X'X (c) Denote H(G; k1, k2,...) by the result of applying the H operator on a symmetric matrix G successively with k- k1,k2..., e.g., H(G; 1,2,3) H(H(H(G; 1);2):3 Update your function H(G; k) constructed in (a) so that it could accommodate the computation of H(G; ki,k2). Your new function call in R shall be in the form of H(G; k), where k can be any subsequence of 1,2,.. .p. Your new function is allowed to use at most one single loop

Answer 1

Solution of the given questions are:

(A) Function which performs the given task.

H <- function(G,k){
    h<-matrix(c(g),nrow=nrow(g))
    for(i in c(1:nrow(g))){
        h[i,]=g[i,]-((g[i,k]*g[k,])/g[k,k])
    }

h[,k]=g[,k]/g[k,k]
    h[k,k]=-1/g[k,k]
h

}

(B)Function call

a<-matrix( rnorm(7*5,mean=0,sd=1), 7, 5)

b<-t(a) %*% a

s=H(b,1)

print(s)

(C) Third part implementation is:

H <- function(g,k){
h<-matrix(c(g),nrow=nrow(g))
for (i in k){
h_1<-apply(h,1,function(x){x=x-((x[i]*h[i,])/h[i,i])})
h_1[,i]=h[,i]/h[i,i]
h_1[i,i]=-1/h[i,i]
h=matrix(c(h_1),nrow=nrow(h_1))
}
h
}

(D)Function call

a<-matrix( rnorm(7*5,mean=0,sd=1), 7, 5)
b<-t(a) %*% a
s_1=H(b,c(3,1,5))
s_2=H(b,c(1,3,5,2,4))
print(s_1)
print(s_2)